Analysis of Simulated Hungarian Income Data

Introduction

This analysis examines simulated income data for Hungary, focusing on the relationships between income and various demographic factors including age, location, occupation, and gender. The dataset is simulated to reflect realistic patterns while maintaining a manageable size for analysis.

Data Simulation

The data was simulated by the data_simulation.R script. The data is available in the hungarian_income_data.csv file.

Important things to note about the generated data:

• Only the 8 most populated cities of Hungary are taken into count weighted by their population. List of cities: Budapest, Debrecen, Szeged, Miskolc, Pécs, Győr, Szombathely, Eger.

• Only the 10 most common occupations are taken into count weighted by their frequency in the workforce. List of occupations: Software Developer, Teacher, Doctor, Sales Representative, Engineer, Accountant, Nurse, Manager, Chef, Driver.

• The age distribution is generated by a beta distribution with parameters \(\alpha = 2\) and \(\beta = 3\) and multiplied by 95 to put the end result in the desired range. The beta distribution with the aforementioned parameters skews the age distribution towards younger ages, which is more realistic.

• There are three groups of people categorized by their age:

  • Underage: each person has a random age at which they start working between 14 and 24.

  • Working age: 19-67

  • Pension age: each person has a random retirement age between 60 and 75.

• Under 18 people have no income.

• Working age people have a regular income based on their age, occupation, city, and gender.

• Pension age people have a pension based on their occupation and city.

• All working age people are considered to be employed.

• The income of a working age man is 20.000 HUF higher than the income of a working age woman in the same occupation, city, and age group.

Demographics Analysis

library(forcats)

data <- data %>%
  mutate(age_group = cut(age, 
                        breaks = seq(0, 100, by = 2), 
                        right = FALSE, 
                        include.lowest = TRUE, 
                        labels = seq(0, 98, by = 2)))

dem_pyramid <- data %>%
  group_by(age_group, gender) %>%
  summarise(count = n(), .groups = 'drop') %>%
  mutate(count = ifelse(gender == "Male", -count, count))

ggplot(dem_pyramid, aes(x = age_group, y = count, fill = gender)) +
  geom_bar(stat = "identity", width = 0.8, color = "black") +
  scale_y_continuous(labels = abs, expand = expansion(mult = c(0.05, 0.05))) +
  scale_fill_manual(values = c("Male" = "#00BFFF", "Female" = "#FF3B3B")) +
  coord_flip() +
  labs(title = "Population Pyramid of Simulated Hungarian Data",
       x = "Age Group",
       y = "Count",
       fill = "Gender") +
  custom_theme +
  theme(legend.position = "top",
        axis.text.y = element_text(size = 10, face = "bold"),
        plot.margin = margin(t = 20, r = 20, b = 20, l = 20))

Data Preprocessing

We employ two steps to clean the data: inter quartile range (IQR) outlier detection, and clustering.

We use IQR outlier detection to reduce the noise of the data.

We employ K-Means clustering to find the three major demographics groups: unemployed/young, working age, retired.

After cleaning the data and finding the three demographic groups, we solely focus on the middle cluster, the working age people, as this study aims to analyze the income of the Hungarian population and it would be nonsensical to analyze unemployed people or retired people as if their pension was a salary.

detect_outliers <- function(x) {
  q1 <- quantile(x, 0.25)
  q3 <- quantile(x, 0.75)
  iqr <- q3 - q1
  lower_bound <- q1 - 1.5 * iqr
  upper_bound <- q3 + 1.5 * iqr
  return(x < lower_bound | x > upper_bound)
}

outliers <- detect_outliers(data$income)
data_clean <- data[!outliers, ]

set.seed(42)
income_age_matrix <- data_clean %>%
  select(income, age) %>%
  scale()

kmeans_result <- kmeans(income_age_matrix, centers = 3, nstart = 25)

data_clean$cluster <- kmeans_result$cluster

cluster_summary <- data_clean %>%
  group_by(cluster) %>%
  summarise(
    mean_income = mean(income),
    .groups = 'drop'
  ) %>%
  arrange(mean_income)

cluster_labels <- c("Working Age", "Pension Age", "Unemployed/Young")
data_clean$income_group <- factor(data_clean$cluster, 
                                 labels = cluster_labels[order(cluster_summary$mean_income)])

ggplot(data_clean, aes(x = age, y = income, color = income_group)) +
  geom_point(alpha = 0.5) +
  scale_color_viridis_d() +
  labs(title = "Age vs Income by Cluster",
       x = "Age",
       y = "Income (HUF)",
       color = "Income Group") +
  custom_theme

data <- data_clean %>%
  filter(income_group == "Working Age")

Descriptive Statistics

summary_stats <- summary(data)
kable(summary_stats, caption = "Summary Statistics of the Dataset (Outliers Removed)") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)

Summary Statistics of the Dataset (Outliers Removed)

	age	city	occupation	gender	income	starting_age	retirement_age	age_group	cluster	income_group
	Min. :14.00	Length:7375	Length:7375	Length:7375	Min. :379808	Min. :14.00	Min. :60.00	30 : 397	Min. :1	Working Age :7375
	1st Qu.:29.00	Class :character	Class :character	Class :character	1st Qu.:529234	1st Qu.:18.00	1st Qu.:65.00	32 : 377	1st Qu.:1	Pension Age : 0
	Median :39.00	Mode :character	Mode :character	Mode :character	Median :579982	Median :19.00	Median :67.00	34 : 377	Median :1	Unemployed/Young: 0
	Mean :40.14	NA	NA	NA	Mean :582750	Mean :18.89	Mean :67.07	28 : 369	Mean :1	NA
	3rd Qu.:50.00	NA	NA	NA	3rd Qu.:632598	3rd Qu.:20.00	3rd Qu.:69.00	42 : 365	3rd Qu.:1	NA
	Max. :73.00	NA	NA	NA	Max. :864318	Max. :24.00	Max. :75.00	40 : 360	Max. :1	NA
	NA	NA	NA	NA	NA	NA	NA	(Other):5130	NA	NA

By the following income distribution plot, we can clearly see that on average a man has a higher income than a woman. This does not yet mean that given equal positions a man earns more money. However, it is indicative that we should further analyze this aspect of the data.

ggplot(data %>% filter(age >= 18), aes(x = income, fill = gender)) +
  geom_density(alpha = 0.6) +
  scale_fill_viridis_d() +
  labs(title = "Income Distribution by Gender",
       subtitle = "(working age only)",
       x = "Income (HUF)",
       y = "Density") +
  custom_theme

The following plot shows how the income is distributed against the age. An important thing to note is that as a person ages, their income increase. However it plateaus after a point, moreover, it even decreases in certain cases.

ggplot(data, aes(x = age, y = income, color = gender)) +
  geom_point(alpha = 0.1, width = 0.2) +
  scale_color_viridis_d() +
  labs(title = "Income Distribution by Age",
       subtitle = "(working age only)",
       x = "Age",
       y = "Income (HUF)") +
  custom_theme

To get a better grasp of how the income distribution is made up, we can split the data by occupation, giving us a new perspective into how certain occupation are more handsomely rewarded. We can see that Software Developers and Doctors have the highest income, compared to Sales Representatives who earn a lower income.

ggplot(data %>% filter(age >= 18), aes(x = reorder(occupation, income, FUN = median), y = income, color = occupation)) +
  geom_boxplot(alpha = 0.7) +
  geom_jitter(alpha = 0.1, width = 0.2) +
  scale_color_viridis_d() +
  coord_flip() +
  labs(title = "Income Distribution by Occupation",
       subtitle = "(working age only)",
       x = "Occupation",
       y = "Income (HUF)") +
  custom_theme

The following plot, like the one before, split the data. However, now we are analyzing how the city in which the person works at contributes to their salary. It is hard not to notice that the average person working in the capital, Budapest, enjoys a higher income compared to other cities.

ggplot(data %>% filter(age >= 18), aes(x = reorder(city, income, FUN = median), y = income, fill = city)) +
  geom_violin(alpha = 0.7) +
  geom_boxplot(width = 0.2, alpha = 0.5) +
  scale_fill_viridis_d() +
  coord_flip() +
  labs(title = "Income Distribution by City",
       subtitle = "(working age only)",
       x = "City",
       y = "Income (HUF)") +
  custom_theme

income_by_category <- data %>%
  filter(age >= 18) %>%
  group_by(occupation, city, gender) %>%
  summarise(
    mean_income = mean(income),
    count = n(),
    .groups = 'drop'
  ) %>%
  arrange(desc(mean_income))

# heatmap
ggplot(income_by_category, aes(x = city, y = occupation, fill = mean_income)) +
  geom_tile() +
  scale_fill_viridis(name = "Mean Income (HUF)") +
  facet_wrap(~gender) +
  labs(title = "Mean Income by Occupation, City, and Gender",
       subtitle = "(working age only)",
       x = "City",
       y = "Occupation") +
  custom_theme +
  theme(axis.text.x = element_text(angle = 45, hjust = 1),
        strip.text = element_text(face = "bold"))

top_earners <- income_by_category %>%
  arrange(desc(mean_income)) %>%
  head(10)

kable(top_earners, 
      caption = "Top 10 Highest Earning Combinations",
      digits = 0) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), 
                full_width = FALSE)

Top 10 Highest Earning Combinations

occupation	city	gender	mean_income	count
Software Developer	Budapest	Male	745832	99
Software Developer	Budapest	Female	722333	100
Doctor	Budapest	Male	720956	61
Doctor	Budapest	Female	696091	77
Software Developer	Debrecen	Male	695662	43
Software Developer	Szeged	Male	694999	30
Manager	Budapest	Male	692519	107
Software Developer	Szeged	Female	684845	43
Engineer	Budapest	Male	675140	133
Software Developer	Debrecen	Female	674727	40

Hypothesis Testing

Parametric Tests

1. Gender Income Difference (F-test + t-test)

Previously we have seen from a plot that a man on averages is payed more than a woman. To test this result we propose a two sample t-test (assuming that both samples come from a normal distribution), where our nulhypothesisis that the mean male income is equal to the mean female income. Before employing a two sample t-test we must first investigate if the variance of the two samples differ significantly, for such we use an F-test.

f_test_result <- var.test(income ~ gender, data = data)
print(f_test_result)

## 
##  F test to compare two variances
## 
## data:  income by gender
## F = 0.99942, num df = 3636, denom df = 3737, p-value = 0.986
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  0.9369317 1.0661025
## sample estimates:
## ratio of variances 
##           0.999418

Given the above results we accept that the variance of the two samples are equal, because we did not find significant proof to state that the variances differ. Thus, we proceed our investigation with the assumption that the variance of the distributions from which the two samples are drown from are equal.

# Test if there's a significant difference in income between genders
t_test_result <- t.test(income ~ gender, data = data, alternative="less", var.equal=TRUE)
print(t_test_result)

## 
##  Two Sample t-test
## 
## data:  income by gender
## t = -10.655, df = 7373, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group Female and group Male is less than 0
## 95 percent confidence interval:
##      -Inf -15335.5
## sample estimates:
## mean in group Female   mean in group Male 
##             573558.0             591693.7

The above results show that the p-value is way below our target of \(0.05\), which means we can state with outmost certainty that we found significant proof that the mean income in the female group is less than the mean income in the male group.

2. Age Distribution Across Cities (7 t-tests)

Living in the capital Budapest, it is apparent to see when visiting other cities in Hungary that there are more young people in Budapest than in the province. This could be caused by many factors, but certainly one is that Budapest has the most and the best universities in Hungary. Thus, many young people from other cities in Hungary, relocate to Budapest to get higher education.

We wish to test if our presumptions are correct, by testing if the mean age across different cities is equal or not. To test this, we employ a t-test, with the assumption that the age in every city corresponds to a normal distribution, where we compare the mean age of every city to that of Budapest.

# compare each city's mean age with Budapest's mean age
budapest_age <- data$age[data$city == "Budapest"]
other_cities <- unique(data$city[data$city != "Budapest"])

t_test_results <- data.frame(
  City = character(),
  t_statistic = numeric(),
  p_value = numeric(),
  mean_diff = numeric(),
  stringsAsFactors = FALSE
)

for (city in other_cities) {
  city_age <- data$age[data$city == city]
  t_test <- t.test(budapest_age, city_age)
  t_test_results <- rbind(t_test_results,
                        data.frame(
                           City = city,
                           t_statistic = t_test$statistic,
                           p_value = t_test$p.value,
                           mean_diff = mean(budapest_age) - mean(city_age)
                         ))
}

kable(t_test_results,
      caption = "T-test Results: Comparing Mean Ages with Budapest",
      digits = 4) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                full_width = FALSE)

T-test Results: Comparing Mean Ages with Budapest

	City	t_statistic	p_value	mean_diff
t	Miskolc	0.5700	0.5688	0.3150
t1	Pécs	-0.9066	0.3648	-0.5335
t2	Debrecen	-0.4749	0.6349	-0.2209
t3	Szeged	-0.9718	0.3313	-0.5160
t4	Eger	1.2779	0.2017	0.7951
t5	Győr	-0.1126	0.9104	-0.0671
t6	Szombathely	0.2632	0.7925	0.1824

The above results show that the p-value for any pair is no way near our target of \(0.05\). Thus we accept the null hypothesis for each test and say that we did not find statistically significant evidence that the mean age of a city is different from the of Budapest.

Moreover, we can visualize the age distribution across cities with a box plot for each city. In the following figure we see that it appears to have no difference on the age distribution which city you live in.

ggplot(data, aes(x = reorder(city, age, FUN = mean), y = age, fill = city)) +
  geom_boxplot(alpha = 0.7) +
  scale_fill_viridis_d() +
  coord_flip() +
  labs(title = "Age Distribution Across Cities",
       x = "City",
       y = "Age") +
  custom_theme

age_by_city <- data %>%
  group_by(city) %>%
  summarise(
    mean_age = mean(age),
    sd_age = sd(age),
    n = n(),
    .groups = 'drop'
  ) %>%
  arrange(desc(mean_age))

kable(age_by_city, 
      caption = "Age Statistics by City",
      digits = 1) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), 
                full_width = FALSE)

Age Statistics by City

city	mean_age	sd_age	n
Pécs	40.6	13.2	619
Szeged	40.6	13.4	836
Debrecen	40.3	13.2	1155
Győr	40.2	13.1	592
Budapest	40.1	13.0	2574
Szombathely	39.9	12.8	397
Miskolc	39.8	13.0	703
Eger	39.3	12.7	499

Non-parametric Tests

1. Gender Income Distribution (Kolmogorov-Smirnov Test)

male_income <- data$income[data$gender == "Male"]
female_income <- data$income[data$gender == "Female"]
ks_test <- ks.test(male_income, female_income)
print(ks_test)

## 
##  Asymptotic two-sample Kolmogorov-Smirnov test
## 
## data:  male_income and female_income
## D = 0.10969, p-value < 2.2e-16
## alternative hypothesis: two-sided

2. Chi-Squared Test for Income Distribution by City

To investigate if there is a significant difference in income between cities we employ a chi-squared test. We first create income categories based on the quantiles of the income distribution: Low (0-25%), Medium-Low (25-50%), Medium-High (50-75%), High (75-100%). Then we apply the chi-squared test to the data. The test will show if the distribution of income categories is independent of city or if there’s a significant association. The visualizations help us understand the nature of these relationships by showing the proportion of each income category within each city.

data <- data %>%
  mutate(income_category = cut(income,
                              breaks = quantile(income, probs = seq(0, 1, 0.25)),
                              labels = c("Low", "Medium-Low", "Medium-High", "High"),
                              include.lowest = TRUE))

city_chi <- chisq.test(table(data$city, data$income_category))
print(city_chi)

## 
##  Pearson's Chi-squared test
## 
## data:  table(data$city, data$income_category)
## X-squared = 2624.1, df = 21, p-value < 2.2e-16

ggplot(data, aes(x = city, fill = income_category)) +
  geom_bar(position = "fill") +
  scale_fill_viridis_d() +
  labs(title = "Income Distribution by City",
       x = "City",
       y = "Proportion",
       fill = "Income Category") +
  custom_theme

The chi-squared test results show a very strong relationship between city and income distribution (p-value < 2.2e-16). This extremely small p-value indicates that we can reject the null hypothesis with very high confidence. In other words, there is a statistically significant association between the city where someone works and their income category. Looking at the visualization, we can observe that Budapest has a notably higher proportion of high-income earners compared to other cities, while other cities such as Eger and Miskolc have the majority of their population composed of low-income earners.

3. Chi-Squared Test for Income Distribution by Occupation

To further explore the components that make up the income distribution, we employ a chi-squared test to test if there is significant association between the occupation and the income. Our null hypothesis is that the distribution of income categories is independent of occupation.

occupation_chi <- chisq.test(table(data$occupation, data$income_category))
print(occupation_chi)

## 
##  Pearson's Chi-squared test
## 
## data:  table(data$occupation, data$income_category)
## X-squared = 2736.5, df = 27, p-value < 2.2e-16

ggplot(data, aes(x = occupation, fill = income_category)) +
  geom_bar(position = "fill") +
  scale_fill_viridis_d() +
  labs(title = "Income Distribution by Occupation",
       x = "Occupation",
       y = "Proportion",
       fill = "Income Category") +
  custom_theme

The chi-squared test results show an extremely strong relationship between occupation and income distribution (p-value < 2.2e-16). This extremely small p-value indicates that we can reject the null hypothesis with very high confidence, meaning there is a statistically significant association between a person’s occupation and their income category.

Looking at the visualization, we can observe that: 1. Software Developers and Doctors have a much higher proportion of high-income earners compared to other occupations 2. Sales Representatives and Drivers tend to have a higher proportion of lower-income categories 3. The income distribution varies significantly across different occupations, highlighting the impact of career choice on earning potential

Regression Analysis

Age vs Income

ggplot(data %>% filter(age >= 18), aes(x = age, y = income, color = gender)) +
  geom_point(alpha = 0.3) +
  geom_smooth(method = "loess", se = TRUE) +
  scale_color_viridis_d() +
  labs(title = "Relationship between Age and Income",
       subtitle = "(working age only)",
       x = "Age",
       y = "Income (HUF)") +
  custom_theme

Multiple Linear Regression

# convert categorical variables to factors
data_working_age <- data %>% filter(age >= 18)
data_working_age$city <- as.factor(data_working_age$city)
data_working_age$occupation <- as.factor(data_working_age$occupation)
data_working_age$gender <- as.factor(data_working_age$gender)

# fit model
model1 <- lm(income ~ age + I(age^2) + city + occupation + gender, data = data_working_age)

model_summary <- summary(model1)
kable(tidy(model_summary), caption = "Multiple Linear Regression Results (Working Age Only)") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)

Multiple Linear Regression Results (Working Age Only)

term	estimate	std.error	statistic	p.value
(Intercept)	434779.5680	3042.221285	142.91517	0
age	8492.2241	146.411115	58.00259	0
I(age^2)	-79.2278	1.733131	-45.71369	0
cityDebrecen	-50118.9464	885.873428	-56.57574	0
cityEger	-100673.3825	1223.767096	-82.26515	0
cityGyőr	-98817.6008	1136.347121	-86.96075	0
cityMiskolc	-99847.5434	1064.213618	-93.82284	0
cityPécs	-98754.3471	1121.335644	-88.06850	0
citySzeged	-49284.8644	997.322532	-49.41718	0
citySzombathely	-99991.5016	1348.646293	-74.14212	0
occupationChef	-42406.4096	1535.934166	-27.60952	0
occupationDoctor	59742.5726	1554.207886	38.43924	0
occupationDriver	-50964.5811	1535.962289	-33.18088	0
occupationEngineer	18051.4651	1251.405236	14.42496	0
occupationManager	37826.6478	1351.616466	27.98623	0
occupationNurse	-32257.7329	1187.823213	-27.15702	0
occupationSales Representative	-60650.2152	1057.574345	-57.34842	0
occupationSoftware Developer	88380.8010	1335.055549	66.20009	0
occupationTeacher	-22511.0613	1128.935083	-19.94008	0
genderMale	18870.6304	582.757427	32.38162	0

par(mfrow = c(2,2))
plot(model1, col = income_palette[1], pch = 19, cex = 0.7)

Polynomial Regression for Age-Income Relationship

# polynomial regression
model2 <- lm(income ~ poly(age, 3), data = data_working_age)

# static plot
ggplot(data_working_age, aes(x = age, y = income)) +
  geom_point(alpha = 0.1, color = income_palette[1]) +
  geom_smooth(method = "lm", formula = y ~ poly(x, 3), 
              color = income_palette[5], fill = income_palette[5], alpha = 0.2) +
  labs(title = "Polynomial Regression: Age vs Income",
       subtitle = "Cubic polynomial fit with confidence interval (Working Age Only)",
       x = "Age",
       y = "Income (HUF)") +
  custom_theme

# model comparison
model_comparison <- data.frame(
  Model = c("Multiple Linear", "Polynomial"),
  R_squared = c(summary(model1)$r.squared, summary(model2)$r.squared),
  Adj_R_squared = c(summary(model1)$adj.r.squared, summary(model2)$adj.r.squared)
)

kable(model_comparison, caption = "Model Comparison (Working Age Only)") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = FALSE)

Model Comparison (Working Age Only)

Model	R_squared	Adj_R_squared
Multiple Linear	0.8851681	0.8848691
Polynomial	0.1541084	0.1537614

Predictions

What is the predicted income of a 35 years old male software developer working in Budapest?

new_data <- data.frame(
  age = 35,
  city = "Budapest",
  occupation = "Software Developer",
  gender = "Male"
)

# Predict income
prediction <- predict(model1, newdata = new_data, interval = "prediction")
print(prediction[1])

## [1] 742204.8